A taxonomy of didactic roles of dynamic visualization in animated mathematics videos

Abstract

In recent years, a multitude of innovative educational videos focusing on mathematics and employing dynamic visualization techniques have been published on YouTube. From a sample of videos, we develop a taxonomy of didactical roles of dynamic visuals and supporting animation techniques. The taxonomy is constructed through an open coding approach, involving an iterative process of defining, analysing and refining the codes. Additionally, we conducted an interview with a video creator about the intended roles of dynamic visuals employed in his animations. The taxonomy encompasses the following categories of roles: connect objects, (co)vary objects, dynamic concept visualization, dynamic process visualization, symbol sense, connect to reality and generalization. These roles are implemented through animation techniques like moving and morphing. The nomenclature of these roles suggests their affiliation with various mathematics education research perspectives, including covariational thinking, concept–process duality, symbol sense and conversion between registers. The taxonomy serves as a foundational framework for further research into the learning outcome and effectiveness of dynamic visualization in educational mathematics videos.

1 Introduction

Informal educational mathematics videos on platforms like YouTube enjoy increasing popularity, exemplified by channels such as Numberphile and 3Blue1Brown, with millions of subscribers. Students use these videos as a learning resource, even if not formally recommended (Trouche et al., 2019; Pepin & Kock, 2021). The videos on these channels present a wide range of innovative dynamic visuals. It is important for our research community to critically examine the visual ingenuity within these videos and establish connections to theoretical perspectives in mathematics education research. Notably, the videos appear not only to engage viewers but also teach and clarify mathematics, if only apparent from self-report through an abundance of positive comments under the videos, such as: ‘To me, this is by far the most intuitive/visible way of explaining the concept. Thank you!’ (linkedout, 2021). However, one may doubt the self-report, and for us, this study is also the starting point for delving further into the effectiveness of dynamic visualization in mathematics education videos. We report on this next step in the second paper of this series on dynamic visualization in mathematics education (Bos & Wigmans, 2023). The current, first paper of the series discusses how online animated video creators employ dynamic visuals to illuminate mathematical subjects.

Previously, scholars have extensively investigated visualization in mathematics education (e.g. Duval, 1999; Arcavi, 2003; Presmeg, 2014). However, the focus has predominantly been on either static or interactive dynamic visualization (e.g. Kidron & Zehavi, 2002; Abrahamson et al., 2014; Chan & Leung, 2014). Additionally, we believe it is highly valuable to examine non-interactive dynamic visuals as employed in animation videos on mathematics¹. So far, we have not encountered published research that explores how creators employ dynamic visualization in such videos and what the learning outcome is. We expect that most education experts would state that learning through interactive visualizations is usually more desirable than through non-interactive visualizations, but nevertheless, since watching videos is such a common practice amongst students, we believe this study is well justified.

In her 2006 paper, Presmeg posed 13 questions concerning visualization in mathematics education (Presmeg, 2006). Amongst them, question 3: ‘What aspects of the use of different types of imagery and visualization are effective in mathematical problem-solving at various levels?’. The current paper attempts to classify different types of dynamic visualization and a follow-up paper delves into their effectivity. Presmeg’s ambitious question 13 ‘What is the structure and what are the components of an overarching theory of visualization for mathematics education?’ remains unanswered to our knowledge, but is on the back of our minds as a long-term objective. Notably, many of Presmeg’s questions still stand in 2024. Since 2000 the research on dynamic visualization in mathematics education has diffused (Schoenherr & Schukajlow, 2023). Researchers in our field have embraced that visualization is just one mode in a spectrum of modes of cognitive activity. Moreover, many have adopted a semiotic perspective focusing on visuals as a particular form of sign; how visual signs are perceived as mediators or parts of activities (Presmeg et al., 2018). Moreover, gestures and body motion gained popularity as objects of study as theories on embodied cognition found their way into mathematics education research. These theories emphasize how the perception of visuals is dynamically coupled with possibilities of acting (Alberto et al., 2022). On a different track, the current paper considers certain visual learning resources, namely animated videos, with a didactical lens. We are interested in the didactical approaches to teaching mathematical topics that are facilitated by dynamic visualization. Hence, the approach in this study is to develop a taxonomy of dynamic visualizations in educational mathematics animation videos based on a medium-sized sample. This taxonomy focuses on the roles of dynamic visuals, with the understanding that our goal is not to achieve comprehensive coverage, as future developments will certainly bring new techniques and ideas for animation.

We build on previous theoretical work on visualization and broaden the perspective on dynamic visuals. In particular, we use the word ‘role’ in line with Duval (1999) and Arcavi (2003) as ‘intended educational function’, representing how video creators, as teachers, intend visuals to contribute to the viewers’ learning process. Duval noted that analysing the role of each visual is an important step in researching different types of visualizations and their effectiveness. The effectiveness of a particular visual can only be determined if one knows the role it serves in the teaching process of the video (Ploetzner et al., 2020). Twenty years ago, in his seminal ESM-paper, Arcavi (2003) described six different roles that visualizations can play in learning and teaching mathematics, e.g. ‘support and illustration of essentially symbolic results’. In this study, we classify the roles of dynamic visuals in animation videos and discuss how these roles are implemented by computer animation techniques, like morphing or moving objects on the screen. We analyse educational animation videos on various mathematical topics through open coding. Additionally, in order to include the perspective of a creator, we interviewed Grant Sanderson, an international authority on animated mathematics videos. He is well known for his videos on YouTube channel 3Blue1Brown and for organizing the yearly mathematics education competition Summer of Math Exposition. He was interviewed about the techniques and roles of dynamic animations in his videos.

2 Theoretical background

2.1. Visualization

The word visualization has various meanings. When used as a verb (visualizing), it can mean creating, using, imagining or interpreting visual imagery. When used as a noun, it is either the process or the product of doing these things (Bishop, 1973). In literature, there is a tendency to embrace this flexibility, leading to varying and lengthy definitions (Phillips et al., 2010). To avoid ambiguity, we will use the word visual to refer to the physical products of the visualization process. Visuals are observable through vision, and, in the context of mathematics, they represent a mathematical object, process or line of reasoning. Presmeg (2006) emphasizes the spatial nature of visuals, meaning that visuals not only include geometric images, but also spatial arrangements of symbols. A mathematical function, for instance, can be visualized by a graph displayed on a computer screen, but also by a formula written on paper. In this study we focus on visuals in animation videos, meaning a dynamic sequence of images on screen delimited in time and place as determined by coherence, e.g. think of a graph changing shape or a simulated moving pendulum.

Besides the differing opinions on the definition of visualization, there exist various perspectives on the role it plays in mathematics education. Arcavi (2003) explains how visualization supports and scaffolds mathematical learning, and he points out that advanced mathematicians rely more on symbolic forms—which we, however, consider visual in themselves. Duval (1999) emphasizes that understanding visuals is not just supportive of learning, but constitutes a learning goal in itself. In particular, he emphasizes that moving between different types of (visual) representations—e.g. connecting a formula to a graph—or ‘conversion’ as Duval (1999) calls it, is fundamental to understanding and problem-solving in mathematics. Both Arcavi and Duval stress that visualization could cause difficulties in learning processes. According to Duval, many errors and misunderstandings can be explained by the inability to convert from one visual representation to another. We think both perspectives are important for dynamic visuals in animation videos: dynamic visuals can support learning goals, and, in particular, foster the conversion between representations.

2.2. Dynamic visualization and animation

The aspect of ‘dynamism’ is essential in this study of visualization. But what does dynamic mean in this context? Presmeg (2006) refers to ‘dynamic imagery’ as images that move or transform. Following this description, dynamic visuals are distinguished from static ones by this additional dimension of movement or transformation. For instance, the translation of a function can be dynamically visualized by continuously moving its graph. The visuals in the videos examined in this study all consist of dynamic visualizations.

The studied videos are animation videos, meaning that the visuals are created using dedicated animation software, rather than captured by camera. In creating such animation videos animators employ animation techniques, including moving and morphing on-screen objects, fading in, changing colour and scaling. So, by a visualization technique, we mean a way to move or transform on-screen objects. This is meant in a technical way, close to the routines and commands used in the dedicated animation software. A visual in an animation video may comprise a combination of animation techniques. In our results, we describe several examples of these combined techniques and how they support a role for mathematical learning.

We believe that visualization plays a special role in mathematics education, in particular, because of the abstract nature of the field. Some authors have described how visualizations can be used to display abstract relations between abstract objects (Duval, 1999; Arcavi, 2003). In their overview book on visualization in education, Phillips et al. (2010) dedicate a section to animation and conceptual change in science education. We posit that additional theory is needed to elucidate the many opportunities of dynamic visualization specific to mathematics education, like conversion between abstract mathematical objects, and hence, we develop a taxonomy tailored to the mathematics context.

Current research is predominantly focused on interactive dynamic visualization, i.e. objects not only move and transform on the screen but are manipulated by the learner or teacher (e.g. Kidron & Zehavi, 2002; Abrahamson et al., 2014; Chan & Leung, 2014; Kohen et al., 2022; Pittalis & Drijvers, 2023). Amongst other results, this line of study has led to a taxonomy of dragging schemes (Arzarello et al., 2002; Baccaglini-Frank & Mariotti, 2010). Our taxonomy may also apply to interactive dynamic visualization, though generally the possibilities and roles will be richer, and therefore warrant an extended taxonomy. The issue of whether interactive dynamic visualization is more effective than dynamic visualization is still under scrutiny (Rolfes et al., 2020), as is the issue of whether static or dynamic visualization is more effective (Ploetzner et al., 2020; Ploetzner et al., 2021; Bos & Renkema, 2022). Our taxonomy allows us to study the effectiveness of (interactive) dynamic visualization in specific roles, which is exactly what Ploetzner et al. (2020) indicate should be the next step in this research direction. The impact on the learning outcome of dynamic visualization for the various roles that we classify will be the subject of the second paper in the series (Bos & Wigmans, 2023).

2.3. Roles of visuals

Arcavi (2003) describes the various roles visualization plays in mathematics education. Six major roles can be distinguished from his article. According to Arcavi, visuals are used to (a) ‘reveal data’ (pp. 219), (b) ‘support and illustrate essentially symbolic results’ (pp. 223), (c) ‘resolve conflict between symbolic solutions and intuitions’ (pp. 223), (d) ‘help us re-engage with and recover conceptual underpinnings’ (pp. 224), (e) provide points of recognition in problem-solving and (f) manipulate expressions more easily. Arcavi bases these general roles on several specific examples from literature. Roles describe the intended function of a visual as it could either be explained by the creator of the visual or derived from the context by an educational expert. We separate the role from the actual effect it may have on learning outcomes; intentions may not always be realized. However, for teachers and educational designers, it is important to be aware of the roles of the videos they design or advise their students. As Duval (1999) writes: ‘They [teachers] must be able to analyse the function that each visualization can perform in the context of a determined activity’ (p. 24). We use Arcavi’s roles for visuals as a starting point and inspiration for our taxonomy. Whilst Arcavi (2003) does not focus on static visuals explicitly, dynamic visuals do not appear in his article. We believe that dynamic visuals offer new opportunities in the form of new roles. For instance, both Arcavi (2003) and Presmeg (2006) warn that static visuals do not match with the sequential nature of teaching, whilst Byrne et al. (1999) already noted that dynamic visualizations are able to illustrate sequential processes. Hence, our taxonomy of roles of dynamic visualization may be able to clarify opportunities of dynamic visualizations that static visualizations do not offer.

The main objective of this study is to provide a taxonomy of non-interactive dynamic visuals as observed in animation videos focusing on mathematics. Specifically, we aim to develop a taxonomy that categorizes the various roles these dynamic visuals fulfill in terms of their intended function in explaining mathematical content. Furthermore, we aim to establish a taxonomy of animation techniques employed in these videos, as these techniques play a crucial role in enabling the realization of the aforementioned roles of dynamic visuals. Consequently, the overarching research question guiding this study can be formulated as follows:

What are the roles of non-interactive dynamic visuals in animated educational mathematics videos?

As we answer this question, we also discuss through what animation techniques these roles are realized.

3 Method

To answer the research question, we viewed a sample of videos that we coded and analysed on applied techniques and roles.

3.1. Video sampling and dividing into fragments

The goal of the sampling process was to select videos with a wide variety of animation techniques and the ways in which these techniques were used. Moreover, we wanted videos with high quality from a technical and educational perspective, with the aim of constructing a taxonomy of high-quality techniques and roles. The subjectivity of the researchers comes into play here, since we used our personal judgement of quality. Another criterion we took into account was the popularity of the video and the channel that it was part of. We accept that this way we might miss a brilliant video that is watched by few. However, our goal is not to have a sample representative for all mathematics animation videos, but a sample that is rich enough to build an interesting taxonomy, reflecting the state of the art. Since the state of the art of visualization is progressing all the time, we accept that our taxonomy should only be seen as a foundational result, open to later updates. Initially, the authors surveyed a large sample of (100+) animated mathematics videos on the video platform YouTube. These videos were found using the search terms ‘mathematics’ and some mathematical topics, like ‘linear algebra’, in combination with terms like ‘animation’ or ‘visualization’. From these videos, an initial sample of four videos was chosen.

The selected videos were divided into fragments that contained one or more distinct visuals. By visual we mean a piece of on-screen visual information coherent over time and space consisting of a set of objects moving or transforming in relation to each other. The unit of coding was the visual, in the sense that to each visual we attached one or more codes for techniques and roles. This resulted in 25–50 visuals that were coded per video.

3.2. Open coding

Every video was coded by at least two coders: the first coder (first author) and one of the two second coders (second and third author). After coding a video, general findings, including new codes and interesting video fragments, were compared in a discussion between the coders to update the shared codebook. Each fragment was coded according to observed animation techniques and interpreted roles of these techniques. Animations not related to mathematical content or without a didactical role, e.g. storytelling animations about the history of mathematics, were not considered. Other elements of the videos, like audio explanations, were also not coded but were considered relevant contextual factors in interpreting the roles of animations.

The initial sample included videos by 3Blue1Brown (2015) and Mathologer (2015), two YouTube channels popular for their videos on mathematics. For the first of these four videos, each coder created their own set of codes. By discussing the similarities and differences in coding, a tentative shared codebook was created. The next of the four videos led to the introduction of new codes, which were compared to existing codes, in line with the constant comparative method (Corbin & Strauss, 1990). When patterns began to emerge, tentative categories for the codes were created.

3.3. Interview

In order to compare our codes and categorization with the view of a creator of such videos, a semi-structured interview with Grant Sanderson, creator of the 3Blue1Brown channel, was conducted. The goal was to see whether the roles we hypothesized would make sense for a creator; and whether the creator recognized his intentions in our descriptions. The interview took place during an online video call. It was recorded and later transcribed and coded, using the shared codebook: coding whenever the conversation turned to an aspect of the taxonomy. The interviewee was shown several fragments of his own videos and was asked to discuss the techniques he used. For each of the techniques, he was asked to explain the role of the animation in the fragment. After responding to the first clip, our coding of the same clip was shared to allow the interviewee to reflect on our descriptions and also to make clear on what level of abstraction and granularity our coding was done. At the end of the interview, the interviewee was asked whether there were animations or animation techniques in his work that were not yet present in our taxonomy.

3.4. Axial coding

After the interview and the initial sample of four videos, seven other videos were selected to extend the codebook. Videos were chosen covering areas of mathematics different from the first four, in case certain techniques and roles turn out to be topic-specific. Similar considerations played a role in the selection of additional YouTube channels to include in the analysis. During this phase of the coding, the codes were further refined and grouped into larger categories. This continued until the codebook was not changed significantly anymore and the coders agreed that theoretical saturation had been reached.

3.5. Testing codes

Once the codebook was considered complete, a first intercoder reliability test was conducted on a new sample of three 5-min-long videos. However, discussing the results we found some additional minor changes to the codebook were needed so that coders would use it more uniformly. Some codes were described more clearly and some others were removed since they were judged to be irrelevant to answering the research question.

After this, a second intercoder reliability test was conducted. All three coders coded a final 5-min video clip. The fragments for this test were 5- to 20-s intervals. Each interval was coded individually by the three coders. Fleiss’ kappa was computed for each code. By weighing codes according to their frequencies, a weighted kappa was then computed (κ = .55). After discussing the differences in coding and individually adjusting the coding, the kappa scores were calculated again (κ = .89). The remaining ambiguity is largely explained by two factors. Firstly, whilst individual codes are defined by distinct characteristics, some combinations of codes can be employed similarly. Secondly, the roles attributed to brief, 1- to 2-s animations may be perceived as negligible.

4 Results and analysis

In total 15 videos were coded—four initial, then seven more and finally four more for intercoder reliability tests—taken from five different YouTube channels: 3Blue1Brown, Mathologer, Mathologer 2, Numberphile and Zach Star. See Table 1 for an overview of the selected videos.

Table 1

Sample of videos in chronological order of analysis

Number & title	Duration coded	Source	Topic/level	Number of views (Sept 2023)
Initial sample
1. The determinant | Chapter 6, Essence of linear algebra	9:15	3Blue1Brown, 2016	Linear algebra Tertiary	3.2 M
2. 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can’t handle?	29:11	Mathologer, 2019b	Algebra Secondary/tertiary	1.9 M
3. Bayes theorem, the geometry of changing beliefs	14:15	3Blue1Brown, 2019b	Probability Secondary/tertiary	3.6 M
4. The essence of calculus	14:27	3Blue1Brown, 2017a	Calculus Secondary/tertiary	8 M
5. Visualising irrationality with triangular squares	4:06	Mathologer, 2018	Number theory Tertiary	0.5 M
6. The secret of parabolic ghosts	5:15	Mathologer, 2019a	Geometry Secondary	0.1 M
7. Differential equations, a tourist’s guide | DE1	7:15	3Blue1Brown, 2019a	Calculus Secondary/tertiary	3.5 M
8. A cute proof with a very dramatic moment:)	0:49	Mathologer, 2021b	Calculus Secondary	0.1 M
9. A cute visual proof with a great punchline	0:56	Mathologer, 2021a	Geometry Secondary	0.02 M
10. Beautiful Trigonometry—Numberphile	5:17	Numberphile, 2020a	Trigonometry Secondary	0.8 M
11. Colouring Knots—Numberphile	5:26	Numberphile, 2020b	Topology Tertiary	0.2 M
Sample for intercoder reliability tests
12. Dear linear algebra students, This is what matrices (and matrix manipulation) really look like	4:18	Zach Star, 2020	Linear algebra Secondary/tertiary	0.9 M
13. Pi hiding in prime regularities	4:53	3Blue1Brown, 2017b	Number theory Secondary/tertiary	2.3 M
14. Visualizing quaternions (4d numbers) with stereographic projection	4:27	3Blue1Brown, 2018	Algebra & Geometry Tertiary	4.1 M
15. The second most beautiful equation and its surprising applications	5:23	Zach Star, 2019	Topology Secondary/tertiary	0.4 M

To give an impression of the coding process, we begin by presenting a vignette, whilst more vignettes will follow later.

4.1. Vignette A taken from video 7 (12:10–12:30)

In this video, differential equations are introduced using the example of a swinging pendulum with air resistance. A red arrow, representing the angular velocity, is visually attached to the end of the pendulum, changing both in direction and magnitude as the pendulum swings. Additionally, the phase space is introduced as a coordinate system with the angle of the pendulum on the horizontal axis and angular velocity on the vertical axis. Whilst a simulation of the swinging pendulum runs, the corresponding position in the phase space is visualized as a dot—along with projections on the two axes—and the dot’s trajectory is traced (see Fig. 1).

Fig. 1

Screenshot of Vignette A. A moving pendulum (top left) and the trace of the coordinates in the phase space of the pendulum’s angle (centre). Image taken from video 7 (see Table 1)

Table 2

Taxonomy of animation techniques: codes and descriptions

Code		Description
(Dis)appear		Introduce objects to or remove objects from the screen (e.g. fade-in)
	Draw or erase	A line or curve is gradually drawn and/or erased
	Trace	The trajectory of an object is traced by visual means (e.g. by drawing the trajectory)
Morph		Gradually change the shape and/or proportions of an object
	Rule-based morph	Transform an object based on content-relevant rules (i.e. a set of mathematical rules or a physics engine)
	Geometric morph	Change the shape of a geometric object (circle, parallelogram, grid, etc.)
	Symbol morph	Transform a symbol into another symbol
	Highlight morph	Change the shape and/or size of highlights, like coloured boxes
	Copy-morph	Transform a copy of an object, whilst the original object is static
	Scale	Change the size of an object in one or more directions
Move		Continuously move an object from one part of the screen to another
	Move	Change the position of an object continuously
	Rotate	Change the orientation of an object
	Rule-based move	Move or transform an object on the basis of content-relevant rules (i.e. a set of mathematical rules or a physics engine)
	Copy-move	Move a copy of an object away from the original
	Flip over	Visualize the reflection of a 2D object by projecting a 3D rotation
	3D perspective change	Visualize a 2D projection of 3D object showing continuously varying perspectives
Small nudge		Attract attention to objects (mainly used for attracting attention)
	Colour change	Change the colour of an object
	Circle-box	Visualize an object in a box, circle, or other shape
	Pulse	Briefly change an object changes size and/or colour
	Wiggle	Briefly shake or wiggle an object
Other		Previous categories do not apply
	Zoom	Zoom in or out

We observed how two objects are moving on-screen simultaneously and in relation to each other: the pendulum and the dot in the phase space. In other words, the pendulum is covarying with the position of the dot in the phase space. Additionally, the arrow also covaries (morphs) with the pendulum and dot. We realized how this dynamic visualization is related to the concept of covariation, well known in mathematics education research (Carlson et al., 2002). After observing similar roles in other sample videos, Covariation became one of our coding categories. The movement of the pendulum in tandem with the evolving trajectory in the phase space and the morphing of the arrow is essential for making a connection between the three.

Another essential aspect of the animation is the sensation of realism that is achieved by accurately simulating the pendulum movement. This aspect allows viewers to connect what is presented on the screen with a real-life situation. The mathematical visual additions, like the arrow, foster the process of horizontally mathematizing the situation (Freudenthal, 1991), i.e. preparing a situation for description by a more mathematical discourse. After observing similar phenomena in other sample videos, Connect to reality was added as role category and Rule-based morph (referring to the laws of physics underlying the dynamic visualization) as a technique. Another technique that was apparent from this animation is Trace, a way for curves or points or other objects to appear on the screen in the wake of a moving object.

4.2. Taxonomies

We present two taxonomies: one of animation techniques and one of didactical roles. The latter we consider our main contribution. A common standard for naming animation techniques seems not to exist, hence we chose to classify the techniques we encountered ourselves. This allows us to specify which techniques are used for which roles. The granularity and detail of the taxonomy of techniques are made subordinate to the taxonomy of didactical roles. For example, even though one could separate various forms of appearing on the screen (fade-in, move-in, suddenly appear), these are not separate categories, since the difference is not important for our discussion of didactical roles. Our taxonomy of animation techniques contains four distinct categories (see Table 2). The first category includes techniques for making objects appear on and disappear from the screen. The second category contains techniques that morph or change the shape of objects. In the third category, ‘movement’, we have broken down the different ways in which objects change position or orientation. Finally, the category of Emphasis refers to animations that are mostly used for visual cues, but sometimes also have a content-specific role.

The taxonomy of roles has five main categories—each containing three to five roles—and two smaller categories (see Table 3). Each dynamic visual in our sample videos was assigned one or more of these roles. Specifically, for the first two categories, codes were often combined. For example, if the role of a visual was to show the covariation in symbols and graphs, three codes were assigned: ‘vary symbols’, ‘vary graphs’ and ‘covary’. Codes from multiple categories were combined, if applicable. A more elaborate description of each category in Table 3 is outlined below. General distinctions that were made early on during this research were reinforced during the interview, as the creator indicated that ‘that taxonomy resonates’.

TABLE 3

Taxonomy of roles of dynamic visuals: codes and descriptions

Code		Description: the animation …
Connect objects (CO)		Visualizes a connection between objects (e.g. relating a formula and a graph)
	Connect to geometric objects	Emphasizes or suggests a connection between geometric objects to other objects
	Connect to symbols	Emphasizes or suggests a connection between symbols and other objects
	Connect to graphs	Emphasizes or suggests a connection between graphs and other objects
	Connect to other	Emphasizes or suggests a connection between objects of types not mentioned above and other objects
(Co)vary objects (CoV)		Shows how objects change, possibly in relation to each other
	Vary geometric objects	Shows changes in a geometric object
	Vary symbols	Shows changes in numbers and other symbolic expressions
	Vary graphs	Shows changes in a graph
	Covary	Relates multiple objects or quantities varying simultaneously or in connection
	Simulate	Visualizes a simulation of a (real-life) system or situation
Dynamic concept visualization (DCV)		Visualizes a concept dynamically
	Dynamic visual for static concept	Enhances or explains a concept not dynamic in nature using movement or transformation
	Map	Depicts a mathematical map dynamically.
	Identify	Shows the similarity or identification of objects.
	Symmetry	Shows/emphasizes symmetries of an object.
	Approximate	Shows an approximation of an object.
Dynamic process visualization (DPV)		Visualizes steps in a line of reasoning or general procedure/process.
	Geometric process or reasoning	Shows geometric steps in a procedure/process or line of reasoning.
	Symbolic process or reasoning	Shows symbolic steps in a procedure/process or line of reasoning (e.g., animated algebraic/arithmetic manipulation).
	(De)compose geometric objects	Combines different geometric objects to show they form a bigger object, or shows how one geometric object can be broken down into smaller parts.
Symbol sense (SS)		Aids in developing symbol sense (cf. Arcavi, 1994)
	Symbol meaning	Helps the viewer understand the meaning of certain symbols within the given context
	Symbolic rearrangement	Highlights connections or similarities in symbolic expressions by reordering symbols.
	Symbolic (de)composition	Combines symbols to form a bigger expression, or breaks down a big expression into separate parts.
Connect to reality (R)		Connects mathematics to real-life situations, including objects and motions.
Generalization (G)		Shows a similarity across cases to deduce a general statement.

4.3. Connect (CO) and covary objects (CoV)

Connecting objects is an essential part of mathematics, for example connecting a number to a line segment, or a graph to a function. Visually, a connection can be suggested by moving different objects close to each other or even making them overlap (Grady, 1997, p. 27). As Sanderson said during the interview: ‘Having one move to the other is … the simplest possible way of really highlighting the logical connection between those.’ Of particular interest is the case of objects represented in different registers, e.g. a geometric object that is connected to a related symbolic expression: the lengths are displayed next to the sides of a rectangle, and then these numbers are moved to the middle of the rectangle where they multiply to compute the area. In the observed animations, the technique of copying an object, e.g. a number, and moving the copy to a different part of the screen was often used to make the connections between objects explicit. Both Duval (1999) and Arcavi (2003) have noted that coordinating between different registers is a challenge in the case of static visualization. In our taxonomy, we emphasize that dynamic visuals instead have a role in fostering these types of connections across registers.

A particular way in which multiple objects on the screen can be visually connected is by using covariation. In the literature on mathematics education, the term ‘covariational reasoning’ refers to ‘mental actions involved in conceiving two quantities as varying in tandem’ (Saldanha & Thompson, 1998; Carlson et al., 2002). In our taxonomy, the role Covary objects of visualizing covarying (quantities associated to) objects is to provoke such covariational reasoning. A main advantage of using dynamic visuals instead of static visuals is the possibility of visualizing continuous variation.

Imagine, for example, a visual of a scaling square with numbers indicating the side length and the area. With static visualization, you can display a series of examples. With dynamic visualization, the numbers can be updated as the square scales. A technique we often observed for visualizing covarying objects, is tracing, e.g. tracing a graph, whilst the process that the graph describes is simulated on screen as well (see also Vignette A). Indeed, interpreting a graph as two covarying quantities is a classic example of covariational reasoning (Saldanha & Thompson, 1998). Showing a simulation whilst drawing a graph showing a related quantity changing over time is a case where the continuous nature of dynamic visualization is highlighted. The interviewed designer, Sanderson, confirmed these intentions: ‘If two things have to move in lockstep, … having those vacillate in sync with each other, I think can be a nice way to … quickly let someone know that two things are connected and dependent on each other.’

4.4. Dynamic concept and process visualization

In our taxonomy, the category dynamic concept visualization applies to dynamic visuals that are intended to illustrate or clarify a concept. There are several codes in this category. Some concepts (like symmetry or map) have an obvious dynamic aspect already in the concept itself, whilst others (like a square) do not, or at least not explicitly. The code dynamic visual for static concept was used when the dynamic element was used to highlight features of an otherwise static concept. For instance, in one of the sample videos, an animation of an ant walking over a knot diagram illustrated how the crossings of string are presented in such diagrams.

We coded many cases where a mathematical map was dynamically visualized (code: map). The best-known static visualization of a map or function is the graph. This works well for maps |$\mathbb{R}\to \mathbb{R},$||${\mathbb{R}}^2\to \mathbb{R},$| or |$\mathbb{R}\to{\mathbb{R}}^2$|⁠, but for higher dimensions, it becomes more complicated, and one sometimes resorts to using colours. In dynamic visuals, time can serve as an extra dimension. ‘Visualizing a function as inputs literally moving [over time] to outputs’ is something Sanderson reported using in his videos. The static equivalent would be to use arrows to visualize the mapping. Replacing the arrows by movement allows one to visualize a map as an action (see Vignette B). We often observed how (isomorphic) objects were identified (code identify) by the techniques of moving them on top of each other (move and rotate), or by morphing one object into the other (morph). This can be seen as a special case of a map. One example of this was an animation of morphing a coffee mug into a torus, to identify both objects. A homotopy like that can be visualized very well dynamically, as time is part of the definition. Another code we found in this category was for the concept of symmetry (this can be seen as a special case of identify: self-identification), often visualized by techniques like (copy-)move or rotate. Sanderson noted that for static visuals ‘symmetry is not your friend’, as the before and after images are the same (the defining trait of symmetry), but ‘actually seeing it move over time makes it completely unambiguous.’

The code approximate refers to the visualizations of a concept defined through a limit or approximation process. For example, think of the concept of the area of a shape defined by covering it with small and smaller squares. The visual then shows dynamically changing configurations of the squares decreasing in size.

Eisenberg & Dreyfus (1991) already noted that static visuals do not match the sequential nature of teaching and proofs. The possibility of adding, moving and transforming objects over time allows dynamic visuals to be more sequential than their static counterparts. Indeed, this is what we observed in many videos and coded in the category dynamic process visualization. Moving objects around, scaling them and drawing lines all showed steps in mathematical processes or procedures. We recognize the duality between concepts and processes (Sfard, 1991; Gray & Tall, 1994), as, e.g. present in the homotopy visualization—a homotopy can be viewed both as a concept and a process. However, we coded according to which aspect was most prominent in the visual.

Our taxonomy distinguishes two types of processes: geometric processes and symbolic processes. Geometric process visualization usually consists of several steps, e.g. drawing a line, indicating a point or folding an object. These steps can be part of a geometric construction or proof. Composing or decomposing geometric objects, e.g. dividing a parallelogram into a rectangle and two congruent triangles, was treated separately as it communicates a different idea. Sanderson: ‘We’re taking a complicated thing and breaking it into smaller [parts].’ Symbolic procedures refer to algebraic or arithmetic manipulations. Examples of this are: moving a term from one part of an equality sign to the other or moving two numbers together to form their sum as a visualization of addition.

4.5. Symbol sense, realism and generalization

The last three categories of the taxonomy relate to three major themes in mathematics education: the use of symbols, the relation of mathematics to reality and the role of generalization. We observed how designers approach these themes through certain types of visualizations.

Arcavi (1994) describes symbol sense as a ‘feel’ for mathematical symbols. This encompasses the ability to recognize when (not) to use symbols, to read and interpret symbols and to engineer symbolic expressions that help in problem-solving (Arcavi, 1994). Here, ‘reading’ means having a ‘gestalt’ view of an expression, whilst also being able to compartmentalize individual terms. Whenever we observed visualizations of symbols and symbolic rearrangements that fostered symbol sense, we attached a code from the symbol sense category and not the code symbolic process or reasoning.Arcavi (1994) does not view the performance of standardized or algorithmic symbolic and computational procedures as contributing to symbol sense. Dynamical visuals that support symbol sense include, e.g. highlighting terms in an equation or arranging terms in an insightful way. In one video multiple derivations were shown, each yielding a formula. Moving these formulas together to form a big result showed how the formula can be broken down into logical chunks. These types of animations were coded as symbolic (de)composition: relating the individual parts to a global visual, or ‘gestalt’, of a formula. Besides movement techniques, highlights and colour changes were also used to relate the parts to the whole. Other visuals were employed to attribute meaning to abstract symbols. In some cases, the animation simply spelled out the meaning in words, e.g. by transforming ‘|$dA$|’ into ‘difference in Area’. Other times, the meaning of parameters was shown by way of covariation, e.g. by varying the slope parameter of a linear formula whilst visualizing its graph. Finally, symbolic rearrangements highlight patterns, similarities, or differences within symbolic expressions (see Vignette C).

By using realistic-looking objects and movement, some visuals related mathematical content to real-world situations (code: connect to reality). This role is usually implemented through the rule-based move-technique code (see Vignette A), which was introduced to highlight that some movements follow underlying rules, like a physics engine, i.e. computer software that simulates physical systems.

Visuals always require specific examples of more general concepts, e.g. visualizing a tangent requires choosing a specific point on a specific curve. The role of generalization is often subtly implemented by dynamic visualization: a general idea can be visualized by moving dynamically (often continuously) between examples. In the example of the tangent line, the point may be moved along the curve, showing how the tangent depends on the choice of point and even the curve can be morphed continuously to show how the concept applied generally to points on smooth curves. Sanderson: ‘The particular one [case] you’re looking at might be kind of arbitrary based on certain arbitrary choices, and seeing the movement around cements that fact.’

The vignettes below give more insight into how we coded visuals. The vignettes were chosen from our sample to illustrate a wide variety of codes. The reader is advised to watch the videos online.

4.6. Vignette B taken fromvideo 1(2:49–2:54)

This video is part of 3Blue1Brown’s series on linear algebra. A recurring theme in these videos is a visualization of a linear transformation. To visualize a linear map of |${\mathbb{R}}^2\to{\mathbb{R}}^2$|⁠, a two-dimensional grid that represents the input is gradually moved to the corresponding output (see Fig. 2). As Sanderson said: ‘It’s a way of showing a function, right? You have inputs and you have outputs, but rather than graphing it—or all the other ways we might see functions—show it in the form of things moving over time: inputs move over to outputs….that’s an idea that every professor of linear algebra wants to convey. You see it written in the textbooks.’ Rather than show two static images, one for the input and one for the output, the animation shows where individual lines and grid points are mapped to, by continuously morphing the input grid to the output. Sanderson: ‘If you just do the static image thing, it can be a little bit confusing. Like what parts of an input correspond to what parts of an output?’. We coded this visual as Dynamic concept visualization: map, using a morph technique.

Fig. 2

Six screenshots of Vignette B. Viewed from left to right, the screenshots show how the scene evolves: the first three images show the transformation of the grid, the square into parallelogram, and the |$A$| into |$3.0\cdot A$|⁠. The final three show the introduction of ‘det’ and the brackets around the matrix and the ‘3.0’ copy/moving from the parallelogram to the equation. Images taken from video 1

The focus of vignette B is on viewing the value of the determinant as the area of the image of the unit square. The yellow square, with an indicated area of |$A$|⁠, transforms into a parallelogram as the grid transforms. Simultaneously, the symbol “|$A$|” morphs into “|$3.0\cdot A$|” (covary). The determinant appears and the 3.0 is copied and moved over to the equation (copy-move), thereby connecting the geometric objects of the grid and parallelogram with the symbols indicating the determinant of a matrix (connect by move). Although the animation is simple, its purpose is very clear. Sanderson’s explanation supports this: ‘Just having one move to the other is like the simplest possible way of, um, really highlighting the logical connection between those.’

4.7. Vignette C taken from video 2 (20:39–21:15)

The next fragment focuses on symbolic manipulation and symbol sense. In the video, the cubic formula is explained and similarities with the quadratic formula are highlighted. In many episodes of the video, algebraic manipulations are dynamically visualized. Most visuals in the video concern routine operations, e.g. literally moving a term to the other side of the equality sign to solve an equation. In other cases, as in this vignette, similar techniques are used to show connections between various symbolic expressions, thus supporting symbol sense.

The clip begins with the perfect cube identity and a cubic equation (see Fig. 3). By combining the middle terms of the cube identity and reordering all the terms, it looks similar to the cubic. Whilst the similarity might not be noticed by every viewer, the connection between the expressions is made explicit in the next step. Extending the highlights of the parameters |$p$| and |$q$| by moving the coloured rectangles (highlight-morph) and spacing out the cubic so that corresponding terms line up, visually connects the equations. The symbols within each of the coloured areas can then be collected and treated as separate equalities.

Fig. 3

Six screenshots of Vignette C. Viewed from left to right, the screenshots show how the scene evolves: the first two images show how the shared |$3\ u\ v$| factors are separated from |$3{u}^2v+3u\ {v}^2$|⁠; the centre images shows these factors merging and the decomposition into |$3 uv\left(u+v\right)$| taking shape. The bottom left image shows that during the rearrangement, the equality sign is moved; also, the vertical bar of the plus sign transforms into a minus sign that is moved, and the brackets and zero symbol fade in. In the bottom right image, the coloured highlights around the parameter are extended to show the similarity with the rearranged identity. Images taken from video 2

Within the context of the video, the first step of combining the middle terms of the cubic identity is portrayed as a procedure that the viewers are invited to perform themselves. We coded this as a symbolic process rather than an example of symbol sense. The rearrangement of symbols in the next step is displayed as a purposeful manipulation working towards a particular solution. Both this rearranging and the line-up of expressions, which allows for the decomposition into several equations, have a role in developing or supporting symbol sense.

4.8. Vignette D taken from video 5 (3:47–4:40)

The final video is about proving the irrationality of certain square roots (see Fig. 4). First, ‘triangular squares’ are introduced as triangles consisting of a square number of small triangles. Using dynamically visualized symbolic manipulation (symbolic process), it is explained that, if |$\sqrt{3}$| were rational, one could find minimal |$A$| and |$B$| such that |${B}^2+{B}^2+{B}^2={A}^2$|⁠. Vignette D shows how treating the equation as a visual problem of fitting three triangular squares into a bigger one leads to a contradiction: rearranging reduces the problem to a smaller, but similar setting, contradicting the minimality.

Fig. 4

Six screenshots of Vignette D. Viewed from left to right, the screenshots show how the scene evolves: the first three images show how the triangles are rearranged from an order representing the equation to a composition with the small triangles at the corners of the big triangle; the final three images show how the triangles are rearranged so that the resulting overlap and empty space creates a smaller version of the previous composition. Images taken from video 5

The process of rearranging triangles is a step in the visual proof, so is coded as a geometric process. It is visualized by moving and rotating the triangles and indicating the overlap with a darker shade. The visual composition before and after rearranging is intentionally similar, to complete the argument. In terms of dynamic visualization, note that the movement allows the viewer to keep track of the triangles, showing the simplicity of the step.

5 Discussion and conclusion

The goal of this study was to establish a taxonomy of the roles of dynamic visualization in animated mathematics videos. Our findings are presented in Table 3, which outlines such a taxonomy for our sample, whilst Table 2 shows the techniques used to realize these roles. The taxonomy can serve both as a framework for analysing mathematics animations and potentially also as a source of inspiration for creators. The taxonomy captures how dynamic visualization presents opportunities in the context of some well-known challenges in mathematics education: e.g. conversion (Duval, 1999; Arcavi, 2003; Presmeg, 2006), covariational reasoning (Carlson et al., 2002, p. 358), symbol sense (Arcavi, 1994) and mathematizing from realistic situations (Freudenthal, 1991).

It is worth noting the differences in perspective between our taxonomy and the roles described by Arcavi (2003). Whilst our investigation focused solely on the roles or intended functions of dynamic visuals, Arcavi includes the roles of the act of visualizing itself, such as imagining or sketching during problem-solving. Nonetheless, there are areas of overlap: dynamic concept visualization aligns with visuals that help ‘recover conceptual underpinnings’ (Arcavi, 2003, p. 224); and dynamic process visualizations often tell what Arcavi calls a ‘visual story’. Dynamic visualization, as opposed to static visualization, opens the door to more complex concept visualizations and to longer stories. The roles of simulating and dynamic visual for static concept both include visuals that ‘reveal data’. In describing the use of visuals to better understand symbolic expressions, Arcavi (2003) emphasizes symbolic ‘solutions’ and ‘results’. Indeed, static visuals show complete results and overviews of the whole solution, whereas dynamic visuals can show the individual solution steps consecutively in time, building up to the solution. Hence, we see how Arcavi’s insights can be complimented with our dynamic perspective.

The reliability of our results is supported by double coding every video and frequently discussing, comparing intermediate results and triangulating our taxonomy through the interview. The intercoder reliability test showed that the taxonomy could in the end be applied unambiguously to a video.

Three interesting aspects of dynamic visualization are not part of this study. Firstly, the effectiveness of dynamic visuals is not addressed in our taxonomy; and it is not a given. Presmeg (1986) noted that dynamic visualization may draw attention to irrelevant details. Whilst cueing animations might help guide a viewer’s attention, motion could also detract attention from the mathematical content. For example, reflecting on the visualization of a linear transformation as a morphing of space, Sanderson noted that: ‘the information of a function doesn’t require steps in between.’ Adding these fictional steps might mislead viewers into thinking the way in which the grid moves is somehow important. Some visuals make more sense to the creator than to the student. As Arcavi (2003) notes, students do not necessarily see the same as the expert. A lot of research has been dedicated to the effectiveness of dynamic visualization compared to static, but recent work by Ploetzner et al. (2021) suggests that effectiveness depends on the role of the visualization. This taxonomy allows us to investigate in the future in more depth in what roles dynamic visualization is indeed more effective than static.

Secondly, the affective function of animation is not covered by our taxonomy. Indeed, many dynamic visualizations have an emotional function. Sanderson: ‘You care about if things feel alive and if they feel like they’re evoking emotion.’ Sometimes dynamical visualizations are only added to invoke awe or create beauty. In 3Blue1Brown videos, an on-screen agent is often present for ‘emotional connection’ as a ‘way of communicating to a learner … when they should have certain emotions’. Dunsworth & Atkinson (2007) showed such pedagogical agents have a positive effect on learning. Other animations that do not have immediate content-related functions, such as those supporting storytelling elements, may have similar benefits.

Finally, the potential roles of dynamic visualization may depend on the visual abilities of the viewer. For example, students with colour blindness will not benefit from nudges provided by colour, and students hearing impairments may benefit from additional visual information, like subtitles or sign language. A future study could explore the didactical roles of visuals in animation videos with a particular focus on inclusive design.

We have two remarks on our sample of videos. Firstly, our sample videos aim at the upper secondary and tertiary levels. At this level viewing videos seems most popular, maybe since students have more independence in shaping their learning process. We believe our taxonomy to be not specific for these levels; however, it would be interesting to see whether videos aimed at the primary level utilize dynamic visualization in alternative roles. Secondly, our sample videos deal with subjects from mathematics. Nevertheless, our taxonomy should apply to any video about subjects that involve mathematics, e.g. physics or economic sciences.

This study provides a taxonomy of the didactical roles of innovative dynamic visualization by YouTube creators. We found how these roles can be described through several theoretical lenses or ideas from mathematics education research. The next step is to study in more depth how the dynamic visuals in the various roles might affect the students’ learning process and outcome, both from a cognitive and an affective perspective (see Bos & Wigmans, 2023).

6 Declarations

Conflict of interest statement

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors have no financial or proprietary interests in any material discussed in this article.

Data Availability

The video data that is analysed was at the time this study was conducted freely available on YouTube. Due to copyright we do not have opportunity guarantee future availability by downloading the videos and making them available elsewhere.

Authors’ contributions

The authors confirm contribution to the paper as follows. Study conception and design: RG and ATV; coding, analysis and interpretation of results: all authors; draft manuscript preparation: ATV; final manuscript preparation: RB. All authors reviewed the results and approved the final version of the manuscript.

Funding

No funding was obtained for this study.

Footnotes

1

Not taking into account that videos do allow an important form of interaction, namely pausing and rewinding (Wetzel & Ludwig, 2021)

Abe ten Voorde obtained a Master’s diploma in Mathematical Sciences in 2020, followed by a Master’s diploma in Science Education and Communication in 2023. He works as a teacher, developer and designer for the Nederlands Mathematisch Instituut, a commercial organization for mathematics education.

Margherita Piroi obtained a Master’s diploma in Mathematics Education in 2019 at the University of Bologna. She is a PhD student at the University of Turin, where she wrote her Ph.D. thesis Teaching and learning eigentheory in University linear algebra: a multifaceted analysis through different theoretical lenses under the supervision of Ferdinando Arzarello.

Rogier Bos is an assistant professor of mathematics education at the Freudenthal Institute, Utrecht University. His research interests include technology in mathematics education to facilitate visual and embodied experiences, in particular aimed at meaning-making, reasoning and problem-solving.

A. Appendix A. References for videos

3Blue1Brown (2015). Home [YouTube Channel]. Retrieved October 6, 2022, from https://www.youtube.com/c/3blue1brown

3Blue1Brown (2016, August 11). The determinant | Chapter 6, Essence of linear algebra [Video]. YouTube. https://www.youtube.com/watch?v = Ip3X9LOh2dk.

3Blue1Brown (2017a, April 18). The essence of calculus [Video]. YouTube. https://www.youtube.com/watch?v = WUvTyaaNkzM.

3Blue1Brown (2017b, May 19). Pi hiding in prime regularities [Video]. YouTube. https://www.youtube.com/watch?v = NaL_Cb42WyY.

3Blue1Brown (2018, September 6). Visualizing quaternions (4d numbers) with stereographic projection [Video]. YouTube. https://www.youtube.com/watch?v = d4EgbgTm0Bg.

3Blue1Brown (2019a, March 31). Differential equations, a tourist’s guide | DE1 [Video]. YouTube. https://www.youtube.com/watch?v = p_di4Zn4wz4.

3Blue1Brown (2019b, December 22). Bayes theorem, the geometry of changing beliefs [Video]. YouTube. https://www.youtube.com/watch?v = HZGCoVF3YvM.

Linkedout (2021). Re: The determinant | Chapter 6, Essence of linear algebra [Video file]. Retrieved from https://www.youtube.com/watch?v = Ip3X9LOh2dk.

Mathologer (2015). Home [YouTube Channel]. Retrieved October 6, 2022, from https://www.youtube.com/c/Mathologer

Mathologer (2018, April 14). Visualising irrationality with triangular squares [Video]. YouTube. https://www.youtube.com/watch?v = yk6wbvNPZW0.

Mathologer (2019a, March 16). The Secret of Parabolic Ghosts [Video]. YouTube. https://www.youtube.com/watch?v = 0UapiTAxMXE.

Mathologer (2019b, August 24). 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can’t handle? [Video]. YouTube. https://www.youtube.com/watch?v = N-KXStupwsc.

Mathologer 2 (2015). Home [YouTube Channel]. Retrieved October 6, 2022, from https://www.youtube.com/c/Mathologer2

Mathologer 2 (2021a, December 17). A cute visual proof with a great punchline (xpost from my Instagram) [Video]. YouTube. https://www.youtube.com/watch?v = r4gOlttnJ_E.

Mathologer 2 (2021b, December 19). A cute proof with a very dramatic moment:) (xpost from Instagram) [Video]. YouTube. https://www.youtube.com/watch?v = SOBz-aFOH2I.

Numberphile (2011). Home [YouTube Channel]. Retrieved October 6, 2022, from https://www.youtube.com/c/numberphile

Numberphile (2020a, June 16). Beautiful Trigonometry—Numberphile [Video]. YouTube. https://www.youtube.com/watch?v = snHKEpCv0Hk.

Numberphile (2020b, October 11). Colouring Knots—Numberphile [Video]. YouTube. https://www.youtube.com/watch?v = W9uVj9rf73E.

Zach Star (2016). Home [YouTube Channel]. Retrieved October 6, 2022, from https://www.youtube.com/c/zachstar

Zach Star (2019, September 13). The second most beautiful equation and its surprising applications [Video]. YouTube. https://www.youtube.com/watch?v = z-GlM7eTFq8.

Zach Star (2020, March 5). Dear linear algebra students, This is what matrices (and matrix manipulation) really look like [Video]. YouTube. https://www.youtube.com/watch?v = 4csuTO7UTMo.